We now proceed in analogy with the arguments in [1]}, [2]}, [3]}.
Since \((|P|_n,|P|_{n-1})\)  is a fiberwise \(|G|\)  equivariant NDR pair for every \(n\ge 1\)  and \(|P| = \operatornamewithlimits{\mathrm {colim}}_n |P|_n\) , we see
that \((|P|,|P|_n)\)  is a fiberwise \(|G|\)  equivariant NDR pair for every \(n\ge 0\)  (Lemma REF ).
For any \(n\ge 0\)  let \(h_n\colon |P|\times _B I\rightarrow |P|\)  and
\(u_n\colon |P|\rightarrow I\)  be a representation of \((|P|,|P|_n)\)  as a fiberwise \(|G|\)  equivariant NDR pair.
Define functions
\(\hat{\rho }_n\colon |P|\rightarrow I\)  for every \(n\ge 1\)  by
\(\hat{\rho }_n(x) = (1 - u_n(x))u_{n-1}(h_n(x,1)).\) 
